\documentclass{article}

\usepackage{../../util}

\title{Large Cardinals}
\author{Cambridge University Mathematical Tripos: Part III}

\begin{document}
\maketitle

\tableofcontentsnewpage{}

\section{Inaccessible cardinals}
\input{01_inaccessible_cardinals.tex}
\section{Measurable cardinals}
\input{02_measurable_cardinals.tex}
\section{Large large cardinals}
\input{03_large_large_cardinals.tex}

\newpage
\section*{Diagram of large cardinal properties}
Under suitable consistency assumptions, large cardinal properties that appear in higher positions on this diagram have strictly higher consistency strength than properties appearing lower down the diagram.

% https://q.uiver.app/#q=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
\[\begin{tikzcd}
	&&& {\kappa \text{ Reinhardt}} & {0=1} \\
	&&& I1 \\
	&&& I3 \\
	&&& {\kappa \text{ supercompact}} \\
	&&& {\kappa \text{ strongly compact}} \\
	{} & {\smash{\begin{tabular}{c}\text{large}\\\text{medium}\end{tabular}}} &&&&& {} \\
	&&& {\kappa \text{ strong}} \\
	&&& {\kappa \text{ surviving}} \\
	&& {\kappa \text{ Ulam}} & {\kappa \text{ measurable}} & {\kappa \text{ real-valued measurable}} \\
	{} & {\smash{\begin{tabular}{c}\text{medium}\\\text{small}\end{tabular}}} &&&&& {} \\
	&&& {\kappa \text{ weakly compact}} \\
	&&& {\kappa \text{ inaccessible}} & {\kappa \text{ weakly inaccessible}} \\
	&&& {\kappa \text{ worldly}}
	\arrow[from=9-4, to=9-5]
	\arrow[from=9-5, to=12-5]
	\arrow[from=9-4, to=11-4]
	\arrow[from=11-4, to=12-4]
	\arrow[from=12-4, to=12-5]
	\arrow[from=12-4, to=13-4]
	\arrow[shift left=2, from=9-4, to=9-3]
	\arrow["{\text{min.}}", shift left=2, from=9-3, to=9-4]
	\arrow[from=8-4, to=9-4]
	\arrow[from=4-4, to=5-4]
	\arrow[from=3-4, to=4-4]
	\arrow[from=2-4, to=3-4]
	\arrow[from=1-4, to=2-4]
	\arrow[tail reversed, from=1-4, to=1-5]
	\arrow[dotted, no head, from=10-1, to=10-7]
	\arrow[dotted, no head, from=6-1, to=6-7]
	\arrow[from=5-4, to=7-4]
	\arrow[from=7-4, to=8-4]
\end{tikzcd}\]

% inacc -/> worldly: Skolem
% W -/> I: reflection KEP
% M -/> W: reflection, same for surv and strong to surv and measurable
% SC -> strong on ES3 46--48, shows how to make strongly compact cardinals embedding cardinals

The `small large cardinals' are usually considered those cardinals consistent with \( \mathrm{V} = \mathrm{L} \), and such large cardinal properties are usually downwards absolute.
Note that \( L \) has no measurable cardinals.
Indeed, if \( \mathrm{V} = \mathrm{L} \) and \( U \) is a \( \kappa \)-complete nonprincipal ultrafilter on \( \kappa \), then the ultrapower embedding \( j_U : L \to M \) must map to an inner model strictly smaller than \( L \), but such an inner model cannot exist.

There are certain large cardinals called \emph{Woodin cardinals} which sit between strong and strongly compact cardinals.
They represent another boundary between sizes of large cardinal axioms, just like measurable cardinals; smaller large cardinals are sometimes called `medium-sized large cardinals', and the others are called `large large cardinals'.
Woodin cardinals are crucial for understanding the connection between large cardinals and infinite games.
We know very little about large cardinal axioms beyond Woodin cardinals.

\end{document}
